My textbook says that if $f (x)=\frac {g (x)}{h (x)} $ is a rational function of $x $ and $h(x)=0$ does not have any real root then $f (x) $ is continous and non monotonic.
$f (x) $ being continous makes sense. But how can we really say that $f (x) $ is going to be non monotonic. Well, to prove that we can try saying that $f (x) $ will always have at least one maxima or minima. But how can we prove that $f'(x)=\frac{h (x)g'(x)-g (x)h'(x)}{(h (x))^2} $ will always have a root?
I have tried to make some graphs of some functions like $f (x)=\frac {x}{x^2+x+1} $ and $f (x)=\frac {x^2+2}{x^2+8} $ and they are indeed non monotonic. But how can we say for a general case?